Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

Authors Marek Filakovský , Tamio-Vesa Nakajima , Jakub Opršal , Gianluca Tasinato , Uli Wagner

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Author Details

Marek Filakovský
  • Masaryk University, Brno, Czech Republic
Tamio-Vesa Nakajima
  • University of Oxford, UK
Jakub Opršal
  • University of Birmingham, UK
Gianluca Tasinato
  • ISTA, Klosterneuburg, Austria
Uli Wagner
  • ISTA, Klosterneuburg, Austria

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Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Problems, reductions and completeness
  • constraint satisfaction problem
  • hypergraph colouring
  • promise problem
  • topological methods


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